An Idealized Hurricane Catastrophe Model

Tropical Cyclones are recurring threats to the densely populated US coast. Post-tropical storm Sandy was an example of their devastating effects. US historical damage since 1900 scaled for population, inflation, and individual wealth show zero trends. Pareto distributions fitted to the tails seasonally aggregated damage offers some insight. Pareto distribution are power-law approximations to other distributions' tails such that Pr{d > D} = Po (Do /D) ^á , where d is damage, D is cumulative-distribution damage, Do is the threshold damage corresponding to probability Po, and á is the Pareto exponent. They can be interpreted as Taylor series approximations to the tails of log-normal that appear to describe hurricane impacts. What is the origin of the distributions' fat tails? Tropical cyclone intensity is thermodynamically limited and their size is limited by the requirement that the Rossby number (the ratio of wind's rotation about the cyclone center to the rotation of the Earth) must be much greater than one. Thus the prominent tail of the damage distribution must stem from the distribution of assets at risk. It is generally accepted that sizes of populated places obey Zipf distributions in which their sizes are inversely proportional to their largest-to-smallest ranking. Zipf distributions are essentially Pareto distributions with unit exponent.

When coastal county population census data is ranked, plotted on a log-log scale and fitted with a Pareto distribution, early census years, such as 1900, produce Pareto exponents close to 1, while 2000 and 2010 have Pareto exponents as large as two. A reason for this discrepancy is that Zipf distributions describe population centers and not necessarily counties. In 1900 each county tended to be dominated by one population center. By the 21st century, counties often contained multiple centers so that the central limit theorem adjusts their distributions toward a normal. This “aggregation effect” is the key to understanding how damage excedance probabilities with Pareto exponents > 1 arise.

To test the hypothesis that the Pareto distribution of US damage is inherited from the distribution of the assets at risk, an idealized hurricane catastrophe model was created. It is easily configured to explore different scenarios and test sensitivities. The model is based upon an idealized virtual country (Zipfistan) where assets that scale as populations of locales that obey Zipf distributions are scattered randomly along a straight coastline. Realizations encompass a specified number of seasons, nominally 100. They can be executed either with unique Zipfistan demographics or all using a common demographic. Within each season the numbers of hurricane landfalls obeys either Poisson or negative binomial distributions. Intensities are uniformly distributed between 33 ms^-1 and a specified Maximum Potential Intensity, nominally 80 ms^-1. Wind profiles obey either Wood-White or Holland parametric models. Vulnerability curves for populated places follow a sigmoid polynomial curve with specified thresholds of initial damage and total destruction. All specified parameters can be changed from case to case or assigned linear time variations to explore model sensitivities. After each realization, damage is ranked and plotted on log-log scales to be fitted with Pareto distributions. After minor model tuning, results yielded Pareto exponents that agreed reasonably well with experience for both individual landfalls (á = 1.14) and seasonal aggregates (á= 1.37). The Zipfistan model is adaptable enough that a gamut of other experiments is possible. Experiments testing changes in maximum potential intensity, landfall rates and vulnerability over decadal time scales are obvious follow-on investigations.